Sat math question
y is greater than or equal to (x-2)^2 + 4
y is less than or equal to -(x-2)^2 + 6
the ordered pair (a,5), where a is a constant, satisfies the system of inequalities above. what is the least possible value of a?

1

Step-by-step explanation:

Sometimes the easiest way to answer a question like this one is to graph it.

The point has to be closest to 1,5. It can also be 1,5

So the answer is a=1

1

Step-by-step explanation:

Let's plug the point into both inequalities.

We will solve both and see what values of they both have in common and then determine the least of that set.

Let's start with the first:

Subtract 4 on both sides:

We are looking for numbers whose squares are less than 1. Those numbers are between -1 and 1.

So this gives me:

Add 2 on all sides:

Now let's solve the other inequality:

Subtract 6 on both sides:

Multiply both sides by -1:

We are already solved this inequality above.

So the solution for is .

The smallest number of that set is 1.

something that characterises a right triangle is the fact that we may use pythagoras' theorem to find any one of its sides given that we have the length of the other two sides. here, we have the length of the hypotenuse, √(87) yd, and one other side, √(23) yd, thus we can easily find the length of the remaining side using pythagoras' theorem as defined below:

c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

thus, given that c = √(87) and a = √(23), we may substitute these into the formula we defined above to get:

c^2 = a^2 + b^2

(√87)^2 = (√23)^2 + b^2 (substitute c = √87 and a = √23 into the equation)

87 = 23 + b^2 (evaluate (√87)^2 = 87 and (√23)^2 = 23 - remember that the square root of a number, squared will equal that very same number)

64 = b^2 (subtract 23 from both sides)

√64 = b (take the square root of both sides)

8 = b (evaluate √64 = 8)

therefor, the length of the unknown leg in the right triangle is 8 yd (first answer).